The following Functional Equation $$f(x+y+xy)=f(x)+f(y)+f(x)f(y)$$ is called Pompeiu Functional Equation. The solution to this equation is the Pompeiu Function given by $$f(x)=M(x+1)-1$$ where M is multiplicative, i.e, $$M(xy)=M(x)M(y)$$ where $x,y\in R$ .
What I am looking for is an operator analogue of $M$, that is, are there any bounded linear operators that satisfy the multiplicative property as defined above? Do such operators exist in literature? If they do, are they elements of ${Z(t)}_{t>0}$ where ${Z(t)}_{t>0}$ denotes $C_0$- semigroups.